A dynamic life cycle analysis for a Defined Contribution pension plan

U NIVERSITY OF T ^WENTE

F INANCIAL E NGINEERING AND M ANAGEMENT

M ASTER THESIS

A dynamic life cycle analysis for a Defined Contribution pension plan

Author:

E.J. Knol

External supervisors:

K. Lee M. Maradona

UT supervisors:

B. Roorda R.A.M.G. Joosten

October 28, 2019

Management summary

Pension is an important part of our social security. Pension funds are under enor- mous pressure and pension cuts seem unavoidable without political interference.

PME and PMT have already announced that cuts in 2020 are very likely. The cur- rent Defined Benefit pension system is under review and the market shifts towards a Defined Contribution pension plan, with more emphasis on individualisation. Such pension plans are still relatively new and therefore more research is needed in this area. This shift increases the importance of life cycle investing. A life cycle in- vestment strategy attempts to determine the most appropriate asset mix for Defined Contribution pension plan participants to balance their risk and return profiles based on the number of years the participants have until retirement. We have found no substantiation for the fact that the current life cycle performs well under the current economic conditions and leads to an optimal pension benefit. We have contributed to the literature because we have compared existing life cycles with optimised linear and dynamic life cycles, while in the current literature one of them is usually taken.

The design of a dynamic life cycle has not yet been evaluated in the Dutch pension context. This explains the relevance of this research and the answer on the following main research question:

How should the life cycle be designed for a Defined Contribution pension plan?

In order to answer this question we have developed a method to analyse Defined Contribution life cycles. We started with capital calculations to gain insight into the capital development during the working period of a participant. This is used to cal- culate the ratio between the accumulated capital and the discounted value of the expected pension benefits, called the coverage ratio. Note that this coverage ratio is not the same as the definition used in a Defined Benefit pension system. The cov- erage ratio serves as an input for the constant relative risk aversion utility function.

The utility is used to compute the certainty equivalents to compare the different life cycle designs. In addition to the assessment framework, we have built a simulation model to model the interest rate and equity returns. We have used the dynamic Nelson-Siegel model in combination with a vector autoregression model to simulate

iii

IV M ANAGEMENT SUMMARY

these interest rates and equity returns. The addition of a Markov regime switching component to the simulation model is of added value because it provided insight in how dynamic life cycles can be designed depending on the state of the economy.

We have showed with the analysis of the traditional, reverse, and constant life cycles that the currently most used life cycle, the traditional life cycle, should not be seen as a guarantee for the optimal pension result. Which life cycle is preferred depends on the risk aversion of a participant. The reverse, constant, and traditional life cycles are preferred for the low, medium, and high risk aversion perception respectively.

This finding suggests that determining the risk aversion of a participant is of great importance.

In the second part of the life cycle analysis we have performed the optimisation with the goal to maximise the average utility by changing the life cycle. First, this is done for linear life cycles. We have found that the optimised linear life cycles result in higher utility values than the three existing life cycles. The shapes of these life cycles are completely different than the existing life cycles. In case a participant has a low risk aversion most capital is allocated to the return portfolio with a slight decrease over time. For the medium risk aversion profile the life cycle starts with an allocation of around zero percent to the return portfolio and increases to almost forty percent to the return portfolio at the end. In case a participant has a high risk aversion then the return portfolio allocation starts at zero percent and increases to almost twenty percent. These results, together with the sensitivity analysis, have showed that the linear life cycle design is highly dependent on the risk aversion coefficient, especially for the low risk aversion profile. This research goes beyond a linear life cycle which is only a function of age. The dynamic life cycle does not necessarily have to be a linear function and is state dependent in order to incorporate the market conditions in deciding the return portfolio allocation. It appeared that adding these two elements to a life cycle result in higher utilities and more certainty, in terms of coverage ratio, compared to a linear life cycle.

Altogether, we have showed that the most used life cycle, the traditional life cycle, is outperformed by other linear and dynamic life cycles. This implies that when a pension fund offers, or wants to offer, a Defined Contribution pension scheme, it should not be taken for granted that the traditional life cycle should be used. In fact, our research have showed that using a dynamic life cycle, which is non-linear and dependent on the market conditions, adds value to the concept of life cycle investing.

This research can be seen as one of the contributions to highlight the added value of

a dynamic life cycle. More research is needed to identify the implications and risks

of using a dynamic life cycle in a Defined Contribution pension plan.

Contents

Management summary iii

List of acronyms ix

1 Introduction 1

1.1 Company profile . . . . 2

1.2 Dutch pension landscape . . . . 3

1.3 Problem analysis . . . . 4

1.4 Research questions . . . . 6

1.5 Research design . . . . 8

1.6 Scope . . . 10

1.7 Report outline . . . 11

2 Assessment framework 13 2.1 Capital calculations . . . 13

2.2 Measures . . . 20

2.3 Conclusion . . . 21

3 Model building 23 3.1 Interest rate and equity returns . . . 23

v

VI C ONTENTS

3.2 Simulation model . . . 26

3.3 Conclusion . . . 37

4 Life cycle analysis 39 4.1 Life cycle designs . . . 39

4.2 Empirical results . . . 41

4.3 Conclusion . . . 44

5 Optimisation life cycle 47 5.1 Optimisation linear life cycle . . . 48

5.2 Optimisation dynamic life cycle . . . 50

5.3 Sensitivity analysis . . . 53

5.4 Conclusion . . . 55

6 Conclusion 57 6.1 Conclusions . . . 57

6.2 Discussion . . . 59

6.3 Limitations . . . 60

6.4 Further research . . . 61

References 63 References . . . 63

Appendices

A Formula sheet capital calculations 65

B Inputs capital calculations 67

C ONTENTS VII

C Mortality table 68

D Cholesky decomposition 69

E Plot simulated yield curves 71

F Matlab codes 72

F.1 Assessment framework . . . 72

F.2 Model building . . . 76

F.3 Optimisation . . . 82

VIII C ONTENTS

List of acronyms

AOW State guaranteed pension CE Certainty Equivalence CR Coverage Ratio

CRRA Constant Relative Risk Aversion CVAR Conditional Value At Risk

DB Defined Benefit DC Defined Contribution DNB The Dutch Bank

DNS Dynamic Nelson-Siegel model ECB European Central Bank

EPB Expected Pension Benefit EURIBOR Euro Interbank Offered Rate FV Future Value

GDP Gross Domestic Product

HICP Harmonised Index of Consumer Prices LC Life Cycle

MSCI Morgan Stanley Capital International NS Nelson-Siegel

OLS Ordinary Least Squares PME Pension fund MetalElektro

ix

X L IST OF ACRONYMS

PMT Pension fund Metaal & Techniek PV Present Value

RQ Research Question

SBK Strategic Investment Framework

VAR Vector Auto Regression

List of Figures

1.1 Problem cluster. . . . . 6

1.2 Research design. . . 10

2.1 Capital calculations. . . 15

2.2 Accruing the premiums. . . 18

2.3 Discounting the benefits. . . 18

3.1 Historical EURIBOR interest rate percentages (Home Finance, 2019). 24 3.2 Historical equity returns. . . 25

3.3 Flowchart simulation model. . . 26

3.4 Factor loadings DNS model γ = 0.0609 (Diebold & Li, 2005). . . 27

3.5 Graphs regime switches. . . 33

4.1 Traditional, constant and reverse life cycle. . . 41

5.1 Optimised linear life cycles. . . 49

5.2 Dynamic life cycle low risk aversion. . . 51

5.3 Dynamic life cycle medium risk aversion. . . 51

5.4 Dynamic life cycle high risk aversion. . . 52

5.5 Optimised linear life cycles using different risk aversion coefficients. . 54

xi

XII LIST OF FIGURES

List of Tables

1.1 Overview Dutch pension system. . . . 3

3.1 Coefficient matrix State 1. . . 32

3.2 Coefficient matrix State 2. . . 32

3.3 Transition probability matrix. . . 33

3.4 Starting values simulation. . . 34

4.1 Coverage ratios given deterministic interest rates and equity returns. . 42

4.2 Certainty equivalents given stochastic interest rates and equity returns. 43 4.3 Statistics of the existing life cycles. . . 44

5.1 Certainty equivalents of the optimised linear life cycles. . . . 50

5.2 Certainty equivalents of the dynamic life cycles. . . 52

5.3 Statistics of the dynamic life cycles with low risk aversion. . . 53

5.4 Statistics of the dynamic life cycles with medium risk aversion. . . . . 53

5.5 Statistics of the dynamic life cycles with high risk aversion. . . 53

5.6 Life cycles with different excess return calibrations. . . 55

D.1 Covariance matrix State 1. . . . 70

D.2 Covariance matrix State 2. . . . 70

xiii

XIV LIST OF TABLES

Chapter 1

Introduction

In essence, a pension contract is a straightforward product. A premium is paid during working years in exchange for a pension benefit after retirement. A pension fund invests all the collected premiums and strives to get a good investment return while taking the risk into account. However, this long-term process is accompanied by uncertainties and difficulties to design a solid and future-proof pension system.

The Dutch pension system is seen as one of the best in the world but is nevertheless currently a subject of the political debate. The current collective pension system is under review and the market shifts more towards an individualised pension plan.

Because such pension plans are still relatively new, there is the need to do research into an individualised pension plan.

Currently, the market shifts from a current collective pension (Defined Benefit) plan towards a more individual pension (Defined Contribution) plan. In a Defined Contri- bution (DC) pension plan the investment strategy is commonly referred to as a life cycle. A life cycle investment strategy attempts to determine the most appropriate asset mix for DC plan participants to balance their risk and return profiles based on the number of years the participants have until retirement. In the traditional life cycle, which is the most used life cycle in a DC pension plan, more capital is allocated to the return portfolio in the beginning of the accumulation phase. This return portfolio is then linearly substituted for a more matching-like portfolio as the retirement date approaches. But is this actually the optimal life cycle? Does this life cycle result in the optimal pension, given the current market conditions and low interest rate en- vironment? The increasing importance of DC plans and their life cycles justify this research.

This opening chapter focusses on the framework surrounding this research. First of all, in Section 1.1 we give a short introduction about the organisation. Subsequently,

1

2 C HAPTER 1. I NTRODUCTION

some background information is provided about the Dutch pension landscape. In Section 1.3 we cover the problem analysis which leads to a problem statement.

Section 1.4 and 1.5 are devoted to the research questions and research design respectively. Thereafter, in Section 1.6 we briefly address the scope of this research.

Finally, we give a rough thesis outline in Section 1.7.

1.1 Company profile

MN is the fiduciary manager for the Dutch manufacturing industry and the maritime sector. In terms of asset under management they are the third largest in the Nether- lands and the largest in the sector. MN manages e135 billion in assets for more than 2 million people and are committed to their future income (MN, n.d.). The client list of MN includes Pension fund Metaal & Techniek (PMT), Pension fund MetalElektro (PME) and Koopvaardij. MN is a large company with close to a thousand employees and several business units.

This research fits to the business unit portfolio management under the Investment Strategy (Dutch: Strategisch Beleggingsbeleid) of the fiduciary advice department.

The core task of fiduciary advice is to empower the members of the pension fund board making the right decisions in the field of portfolio management through good research which is then translated into policy advice and product development. This department is also responsible for writing investment strategies and mandates for implementation.

A pension fund board is responsible for the investment policy and makes use of supporting parties for advice and implementation. Central to the MN approach is a modern and effective investment framework. In 2015 this investment framework has been formalised into the ”Strategisch Beleggingskader” (SBK). In the framework the objective of the pension fund has been stated and how it wants to achieve the goal.

A unique aspect of this approach is that MN prepares the SBK for all clients in a document in close consultation with the board. It forms the basis for the role as a fiduciary manager.

One of the ambitions of the fiduciary advice business unit is to be able to respond to

the changing pension environment. This ambition serves as the perfect starting point

for research about how the investment policy can be improved given the changes in

the pension system. In the following section we elaborate on the problem that MN is

facing.

1.2. D UTCH PENSION LANDSCAPE 3

1.2 Dutch pension landscape

After seven years the Dutch pension system is again the number one pension sys- tem in the world according to the Global Pension Index 2018 (Mercer, 2018). Since 2009 Mercer compares the quality of pension systems over thirty countries world- wide and is based on three basic elements: adequacy, future-proofing, and integrity.

This ranking indicates that the Dutch pension system is doing really well.

The three pillars

The Dutch pension system consists of three pillars. The first pillar concerns the state-guaranteed pension (AOW) which was introduced in 1957. It is the basic in- come and everyone who lives or works in the Netherlands will receive AOW as soon as the legal retirement age has been reached. The first pillar is financed through a pay-as-you-go system. This means that the working population pays the cost of the AOW of the current pensioners. The second pillar is a collective pension system organised around a specific industry/company. This is funded by the premiums that people have invested in the past, plus the return on it. If the employer has such a supplementary pension scheme, retired employees will receive an additional bene- fit on top of the AOW. The third pillar concerns the individual pension products. In particular, employees in sectors without a pension scheme and self-employed make use of this pillar. Our research focusses on the second pillar of the Dutch pension system.

First pillar State pension Pay-as-you-go Second pillar Occupational pension Funded

Third pillar Individual pension Funded

Table 1.1: Overview Dutch pension system.

Defined Benefit and Defined Contribution

In the current situation the second pillar of the Dutch pension industry offers differ- ent kinds of pension contracts. The majority of the sector in the Netherlands uses the Defined Benefit (DB) pension system. With a DB pension system, the pension payment is guaranteed. A partial entitlement of the pension benefit is accrued for each active year of service, which is based on a percentage of the average salary.

The pension fund then sets the investment policy that is suitable to fulfil the guar-

anteed pension pay-outs. This policy is applied collectively and is the same for all

participants. Several tools are available for the pension board to adjust the financial

position of the fund. These tools include the premiums paid by the active members

(premium policy) and the inflation indexation that applies to all participants (indexa-

4 C HAPTER 1. I NTRODUCTION

tion policy), with a negative indexation as the ultimate variant. The viability of a DB pension system is questionable and is one of the reasons why the current pension system is under discussion.

A Defined Contribution (DC) pension plan is a bit more straightforward. In a DC, the employee and employer contributions are invested on behalf of the employee (Hull, 2015). However, in a DC it is not known in advance what the exact pension benefit will be after retirement. Of course, this depends on the amount contributed but also on the growth of the capital throughout the years. Once the employee retires, the capital can be converted to a lifetime annuity. Because the pension benefit highly depends on the returns it increases the importance of the investments. This is where the life cycle comes into play. This is an investment policy which is often based on a function of the age of the (active) participant.

The majority of the sector in the Netherlands uses the DB pension system. However, numerous alternatives are the subject of discussion in politics and the pension in- dustry. One of the alternatives is a personal pension account with a collective buffer.

This alternative can be seen as a hybrid between a DB and DC.

An important difference between DB and DC is the fact that the latter is a more individual kind of pension system. This can be seen as an individual employee account where the pension benefit is calculated based only on the funds on that account. This is in contrast with a DB pension system where there are no such individual accounts. The contributions are pooled and invested and the pension benefits are paid from the pooled capital. Another big difference is the way the risks are borne. With a DC plan the risk is fully carried by the employee because the pension benefit directly correlates with the total amount of capital of the individual fund. However, an advantage of a DC system is the flexibility. It can be adjusted based on personal characteristics which is not possible with a DB pension system.

1.3 Problem analysis

Although the Dutch pension system is recognised as one of the best internation-

ally, some shortcomings have become increasingly visible in recent years. These

are particularly related to the overarching themes such as the transparency of and

the trust in the system which are under pressure mainly due to the current market

conditions. The historically low interest rate is a good example for the changed cir-

cumstances compared to the past. Does the current (DB) pension system still work

under these circumstances?

1.3. P ROBLEM ANALYSIS 5

Because of the increasing social and political pressure, MN is interested in the ques- tion whether a hybrid pension system, which incorporates good elements of both the DC and DB systems, can meet the objectives. Right now the scope of this question is still quite broad and therefore it is necessary to narrow it down. Based on the preferences of MN, our research zooms in on the life cycle of the DC system. Ques- tioning the current life cycle is enhanced by literature pointing out that more research needs to be done to improve the current life cycle. There are already some stud- ies that evaluated life cycle designs of a DC pension plan, but they are somewhat outdated and therefore not representative for the currently low interest rate environ- ment, for example the study of Blake et al. (2001). However, these papers did not investigate a dynamic life cycle with the same definition as in this study. Next to that, Basu et al. (2011) stated that the LC can be counterproductive when moving away from stocks to low-return assets just when the size of the contributions are growing larger. They tested a dynamic life cycle where it is only allowed to switch between stocks and bonds during the last ten or twenty years. So, our study differs from their research in the definition of the dynamic life cycle. Another article stated that there is room for added value for the one-size-fits-all LC to incorporate classes of investors characteristics such as risk attitude and income (EDHEC-Risk Institute, 2011). In addition to that, the current LC does not incorporate investment results that are very dependent on market behaviour (Arnott et al., 2013). This means that there is no feedback or performance check built in the investment strategy which can have influ- ence on the remaining part of the LC. Poterba et al. (2006) found that the distribution of retirement wealth associated with typical life cycle investment strategies is similar to that from an age-invariant asset allocation strategy. They stated that it might be useful to compare the optimal life cycles with the existing ones. This literature study shows that there is indeed room for improvement and could be seen as an invitation to join the research about the life cycle.

Is it possible to add elements to the life cycle, which is now often a function of age,

so that there will be feedback between the investment policy and the desired end

goal? Is it also possible to include multiple decision factors such as the current level,

remaining life cycle, desired risk, etc.? An interesting question may also be how

the investment policy will be adjusted if a participant has accumulated capital that is

above the final goal (for example 70% of the average salary in accordance with the

current DB ambition) while the participant is only 54 years old. Then you could, for

example, take less risk in the remaining time. Figure 1.1 gives an overview of the

problems related to the life cycle.

6 C HAPTER 1. I NTRODUCTION

Figure 1.1: Problem cluster.

1.4 Research questions

At the time of writing this report, the Dutch pension system is still under review. As the third biggest fiduciary manager in the Netherlands it is important for MN to build a sustainable pension plan, especially given the changing pension landscape.

One of the aspects being discussed in the review is the shift from a Defined Benefit pension plan towards a Defined Contribution pension plan. This trend stems from the more flexible labour market, longevity and ageing population. In addition to that, low interest rate environments have put more pressure to the coverage ratio (CR) of the majority of Defined Benefit schemes in the Netherlands. With the sustainability of the current DB plan being questioned and the trust in the system diminishing, there is an increasing pressure to look into another system such as Defined Contribution plans.

The shift from DB plans to DC plans presents a new challenge for fiduciary man- agers such as MN to advise the pension board how to best manage the retirement assets. In addition to the policy regarding employees contribution, the investment strategy/asset allocation decisions play an important role in determining the pension outcome. The increasing importance of DC plans and its life cycle justifies the need for this research.

The main objective of our research is to investigate the optimal investment strate-

gy/life cycle in a DC plan. The investment strategy of a DC plan (life cycle) involves

allocating the accumulated wealth/assets to equity-like assets (return portfolio) and

1.4. R ESEARCH QUESTIONS 7

bond-like assets (matching portfolio) according to a certain fixed glide path. Cur- rently, this glide path is a function of the participant’s age. To achieve the research objective, we have formulated several research questions. These serve as the ba- sis, guideline and structure for this thesis. We define the main research question as follows:

How should the life cycle be designed for a Defined Contribution pension plan?

An important aspect here is the question when the life cycle is optimal. On one hand, it is clear that a higher pension benefit is better than a lower pension benefit. On the other hand, more certainty in terms of the pension payment is better than less certainty. The problem is that these two outcomes are often substitutes: a higher benefit is usually accompanied by more uncertainty. Therefore, it is necessary to look at the trade-off between risk and return to see which life cycle offers the best result. The main research question is broken down into two research questions.

These form collectively an answer to the main research question.

Research question 1

Which life cycle design offers the best risk-return trade-off given a certain risk- aversion level of the participant and stochastic interest rates and equity returns?

In this step of the research we introduce a set of scenarios for interest rates and

equity returns. The traditional life cycle follows a fixed glide path in which more is

allocated to the return portfolio in the beginning of the accumulation path. This re-

turn portfolio is then substituted for a more matching-like portfolio as the retirement

date approaches. Most of the current DC plans adopt this traditional life cycle as

opposed to a reverse glide path and a constant mix. Although several plans vary

slightly in terms of rates at which the return portfolio is reduced according to the

risk aversion level of the participant, the implementation of a reverse glide path and

a constant mix life cycle is minor. Each life cycle produces a wealth distribution at

the retirement date. The expected pension payout can then be calculated from this

result. The traditional life cycle is the most used strategy. Therefore, in an environ-

ment where interest rates and equity returns are stochastic, we expect that a glide

path with a decreasing return portfolio over the accumulation phase yields superior

pension benefit at the given retirement date in comparison with a reverse glide path

and a constant allocation. No distinction is made between the risk aversion of the

participants and therefore we expect that the life cycle preference is independent

from the risk aversion level of the participant.

8 C HAPTER 1. I NTRODUCTION

Research question 2

What is the impact of adjusting the return versus matching portfolio based on the target pension benefit throughout the working life of the participant?

In this next step of the analysis we propose a new method. Instead of looking at the age of the participant (or the time to retirement) as an anchor point for determining the asset mix, the proposed life cycle defines the asset mix based on the extent to which the target pension benefit is achieved. At each point in time (for example each year) the value of the pension contribution is evaluated against the value of accruals at retirement. As the ratio of the contribution to accruals is higher (i.e. the pension contribution matches the pension payout) a bigger portion of pension benefit is secured by allocating more to the matching portfolio. This is done irrespective of the age of the participant at that point. Based on this idea, we expect that a dynamic life cycle, in which the allocation between return and matching portfolio is managed against the target pension benefit throughout the accumulation phase, generates a better pension result than the traditional life cycle in which the allocation is defined only based on the age of the participant.

Both research questions are related to the design of a life cycle for a DC plan and contribute in their own way to answering the main research question. In the next section we elaborate on the research design.

1.5 Research design

The research design serves as a roadmap for the entire process to achieve the re- search objective. Here, we break down the research questions into smaller steps.

We conduct the analysis through an iterative process, each time with small adjust- ments. In this way the impact of the incremental changes in the analysis can be isolated. In this research we explain and clearly state all assumptions for replication purpose. The main methods and data sources we use are literature studies, the MN pension database, extern data portals such as Bloomberg and the MN employ- ees. Knowledge about the pension industry is gathered through discussions and conversations with MN specialists. In the following paragraphs we elaborate on the different steps that we take to answer the research questions.

The focus of the first step is to understand how to assess the trade-off between risk

and return. There are couple of requirements to be able to assess the different life

cycles. These are related to the capital calculations, utility function, and certainty

equivalent. First, we discuss the capital calculations because pension contributions

1.5. R ESEARCH DESIGN 9

are made during the participant’s working years in exchange for a pension benefit after retirement. Therefore, it is fundamental to do the capital calculations to see how the total accumulated capital will change over time. Next to that, we use the Constant Relative Risk Aversion (CRRA) utility function which is one of the most commonly used utility functions in the pension industry (Yue, 2014) and is often used as an evaluation measure in the literate on dynamic asset allocation. The capital calculations, the CRRA utility function, and the certainty equivalents form the basis for the analysis of life cycle design. In order to conduct this analysis, data from the MN database is used as input. This includes the mortality table, career path percentages, and pension contribution table.

The goal of the second step is to be able to test the life cycle designs under different economic circumstances. We need to generate stochastic interest rates and equity returns to create a more realistic view of the performance of the life cycles. We do some literature study to gain knowledge about different models such as Vasicek, Nelson-Siegel (NS), Heath-Jarrow-Morton framework (HJM), Geometric Brownian Motion (GBM) and Markov regime. Given the scope of this research, we keep the scenario generation relatively simple. This also reduces the dependency on large amount of data. We gather the input data for scenario generation, for example data about swap rates and indices, using Bloomberg.

We use the results of the previous two steps in the third step to analyse three existing unidirectional life cycles. Once we have modelled the capital calculations, utility function, interest rates and equity returns, we test different life cycles to answer Research Question 1. These are referred as the constant, traditional and reverse life cycles, which we discuss in more detail in Chapter 4.

In the last step in this research we perform the dynamic life cycle optimisation. While in step three the design of the life cycle is constant and defined at the beginning of the accumulation phase, in this step we adjust the life cycle design along the way based on the projected pension payout. So, what is the impact of this continuous ad- justment of the life cycle design to the pension benefit at retirement date? The effect of using a dynamic asset allocation approach (dynamic glide path) to incorporate elements such as current level, remaining life cycle, and desired risk is investigated.

However, the scope of this research is limited to determining the optimal allocation

to the matching and return portfolio. The purpose of the matching portfolio is to

hedge away interest rate risks as effectively as possible. The interest rate risk is

the risk that the value of these pension liabilities rises faster than the value of the

total assets. The goal of the return portfolio is to generate returns above the interest

rate based on an optimal risk-return trade-off. We do not investigate which specific

financial products should be used to get the corresponding asset allocation mix.

10 C HAPTER 1. I NTRODUCTION

To summarise, we execute several steps to be able to answer the main research question. These steps together serve as a roadmap for this research. This can be represented in the following flowchart:

Figure 1.2: Research design.

1.6 Scope

Naturally, in every research it is important to determine the scope. First of all, our

research is based on the Dutch pension system and hence not always applicable

to pension systems from other countries. If someone wants to reproduce this re-

search, one should think carefully about the underlying pension system which could

have different assumptions and legal requirements. In addition, our research will

not go into detail about the personal characteristics. This means that personal life

events, which could have an impact on the pension, will not be incorporated. An-

other decision is related to the retirement age. It is impossible to predict what kind

of regulatory changes will happen in the future. Therefore, we use the retirement

age based on the current regulation. Also, our research does not take into account

the changes in demographics. Finally, we do not take taxes, transactions costs,

and leverage into account. All this decreases the complexity and narrows down the

scope of this research.

1.7. R EPORT OUTLINE 11

1.7 Report outline

From the previous sections, the thesis structure can be derived. In Chapter 2 we

discuss the framework to assess the trade-off between risk and return. The assess-

ment framework is built on the basis of two features, the capital calculations and the

utility function. We explain the interest rate and equity return model step by step in

Chapter 3. This is done to be able to assess the different life cycles with stochastic

interest rates and equity returns. In Chapter 4 we test three existing unidirectional

life cycle designs to get a first impression of their performance. In addition, we an-

swer Research Question 1. In Chapter 5 we elaborate on the dynamic life cycle

design to be able to answer Research Question 2. We use all findings to answer the

main research question in Chapter 6.

12 C HAPTER 1. I NTRODUCTION

Chapter 2

Assessment framework

As described in the first chapter our research consists of two research questions with the main objective to see whether a dynamic life cycle outperforms the current unidirectional life cycle. In order to answer the research questions we discuss the mechanism to capture risk-return trade-off first. The goal of this chapter is to come up with a risk-return measure to evaluate each of the life cycle design and to quantify the effect of changing the asset allocation mix. First, it is necessary to understand how the pension payout is calculated and what the pension contributions should be that the participant has to pay to achieve this pension payout objective throughout the accumulation phase. This is referred to as the capital calculations. In this calcu- lation the target pension payout at retirement date is set. We calculate the present value (PV) of the life-long benefit entitlements that are accrued by discounting the annual payout with the interest rates. The annual pension contribution (premium) is then calculated in such a way that at the target retirement date the future value (FV) of these premiums matches the present value of the annual pension payout. The same interest rate structure that is used to discount the pension payouts is used to calculate the future value of the contributions. The next step is related to the utility function. The risk aversion coefficient has to be determined in order to assess the life cycle using the utility function. Finally, the utility serves as an input in the cer- tainty equivalent (CE) calculation. We discuss all steps in more detail in the following sections.

2.1 Capital calculations

The participants pay pension contribution (premium) during their working years in exchange for a pension benefit after retirement. Currently, the retirement age is part

13

14 C HAPTER 2. A SSESSMENT FRAMEWORK

of the political discussion, so there is some uncertainty about what the retirement age will be in the future. For the purpose of this research we have assumed that the pension age is fixed. We have used the following inputs/assumptions in the capital calculations:

• The starting age for the wealth accumulation period is at the beginning of 25.

• The retirement age is at the beginning of 67.

• The last evaluated age is 103.

• The starting annual salary is e27,000.

• The franchise is e15,304.

• The payouts are at the beginning of a year (primo).

• The inflation correction is 0.5% per year.

• The career path is given as input.

• The mortality table is given as input.

• The pension accrual is 1.875% per year.

• The spot interest rate as of 29-3-2019 is used.

• Simulated interest rate and equity returns are used.

• The expected inflation term structure as of end March 2019 is used.

• No life events, like divorces or promotions, are incorporated.

• Partner pension is not taken into account.

First, the capital calculation is applied to a deterministic scenario based on the spot interest rate as of end March 2019. We discuss the use of stochastic scenarios in the next chapter. In each scenario the interest rates and equity returns are defined.

Figure 2.1 gives an overview of the annual capital growth based on the previously

mentioned inputs. In the paragraphs below the figure we explain all calculation

steps. The specific formulas can be found in Appendix A.

2.1. C APITAL CALCULATIONS 15

Figure 2.1: Capital calculations.

The first column is the age of the participant. We have calculated the capital on an annual basis starting with the year when the participant is 25 years old until he/she reaches the age of 103. We have chosen this lifetime range to accommodate the mortality table input; at the age of 103 the life expectancy is close to zero. The pension benefit is paid out at the beginning of the year. The retirement age kicks in when the participant reaches 67 (beginning of the year).

The second column contains the yearly salary information. Recall from the assump-

16 C HAPTER 2. A SSESSMENT FRAMEWORK

tions that the starting salary is e27,000. At any year that the participant is younger than the retirement age (67) he/she earns a salary. The amount of salary changes over time depending on the inflation and the career path. We have assumed that the salary grows with the inflation. The career path shows the percentages with which the salary grows compared to the previous year as a result of a participant’s career.

The next column is the franchise computation. This is the part of the salary on which no pension is accrued and therefore no pension contribution (premium) is paid. It also depends on the starting value and is also effected by the inflation.

Next, the pension base (Dutch: pensioengrondslag) is calculated by deducting the franchise from the salary. The pension base is the part of the salary on which pension is accrued and therefore pension contribution is paid.

The next step is to calculate the pension contribution (premium) by multiplying the pension base with the accrual percentage. As you can see in Figure 2.1, premium is only paid when the participant is 66 years old or younger. In addition to the returns on investment, the inflow of premiums is an important source of capital. The premium policy is one of the management tools that a pension board can use if necessary. In our research, however, we have not used the premium as a steering tool. We have used a fixed contribution table, which can be found in Appendix B.

The contribution percentages are based on the accrual percentage of 1.875% per year and the spot interest rate of March 31, 2019. In other words, this pension contribution must be paid to build up 1.875% pension per year, assuming all capital is allocated to the matching portfolio. We have found that this results in a pension annuity of e33,707.68 and is used as the pension ambition in our research. This stylised framework offers a clearer insight into the influence of the life cycle designs on the pension result.

In the sixth column the pension benefit is calculated. Note that the participant only receives the benefit when he/she is 67 years or older. The pension benefit depends on the total accumulated capital so far and on the annuity factor (Dutch: koopsom).

In this research an annuity factor is an one-time payment to buy a pension entitle- ment (Pensioen.com, n.d.). The annuity factor represents the amount of money that is now needed to be able to buy an annual of one euro pension entitlement which is distributed from retirement age until the participant dies.

The next column contains information about the annual returns (also called EUR return). The return depends on the accumulated capital so far and the way the capital is invested. The investment policy is defined based on the chosen life cycle.

A life cycle is used to determine how much capital is invested in the matching and

2.1. C APITAL CALCULATIONS 17

how much in the return portfolio. In our research the matching portfolio consist of interest rate investments and the return portfolio consist of equities. Therefore, the total annual returns depend on the interest rates and the equity returns.

The last column is the final step to calculate the total accumulated capital. This is called the capital ultimo. It depends on the accumulated capital up to the previ- ous year, the contribution in the current year, the pension benefit paid out this year (if any), and the total annual return on investment. We have used this capital to calculate the pension benefit because it incorporates everything such as premium, pension benefit, and returns from the investment portfolio.

Annuity factor

As mentioned before, we have used the annuity factor to determine the pension ben- efit based on the accumulated capital. The necessary inputs to calculate the annuity factors are age, mortality rate, and spot interest rates. A mortality table shows, for every age, what the probability is that a person of that age will die. The mortality table can be found in Appendix C. By using the mortality table, it is possible to cal- culate the conditional probability of survival at a certain age. The probability that someone is still alive up until a certain age, multiplied by one minus the probability that a person will die at that age, gives the conditional probability of survival. In other words, it is the probability that a participant will survive given that the partic- ipant survived up to now. We have done this for every year, ranging from 25 until 103 years old, assuming that the participant is now 25 years old. These probabilities are needed to generate the expected pension payout table with two dimensions, the current age of the participant and the time horizon. Given a certain age the expected pension payout is calculated. This is done by multiplying the e1 pension entitlement per year (annuity) with the probability that the participant is still alive each year. So, for example given that a person is now 60 years old (t = 35) and the horizon is 7 years (h = 7 ), what is the expect pension payout? We have done this for every age and horizon to generate the expected pension payout table. The expected pension benefit (EPB), dependent on the current age and horizon, is given by:

EPB _t,h =







e1 × ^P _P

, t + h ≥ 67 0 , t + h < 67

, (2.1)

where P is the probability of survival at a current age t ∈ {25, 26, ..., 103} and horizon h ∈ {0, Z ⁺ }.

Once the expected pension benefit table is generated it should be discounted. We

have used the spot interest rate to discount the expected pension benefit. The

18 C HAPTER 2. A SSESSMENT FRAMEWORK

formula to calculate the annuity factor A is given by:

A _t = EP B _t,0 +

T

X

h=1

EP B t,h

(1 + r _h ) ^h , (2.2)

where r h is the spot interest rate with a horizon h.

Now that these annuity factors are calculated we can use them to compute the pension benefit given the accumulated capital so far. This step serves as an input for the capital calculations.

Discounting cash flows

As mentioned in the previous section, the payments of the pension contribution and the pension benefit take place at different moments in time. This means that we need to accrue the contribution payments and to discount the pension benefits to be able to fairly compare the available money and liabilities at the retirement age. This gives an indication about the solvency of the pension fund at the time the participant retires. Figure 2.2 and Figure 2.3 give a schematic overview how the different cash flows are accrued and discounted.

Figure 2.2: Accruing the premiums. Figure 2.3: Discounting the benefits.

The premiums are paid annually and therefore the number of years until retirement decreases annually. This means that the premium paid by a participant at the age of 25 is invested for a period of 42 years while the premium paid at the age of 26 is invested for a period of 41 years and so on.

The discount rate is a critical input parameter for the outcome of the present value

and future value calculations. In the stochastic analysis we have used the simulated

yield curves to accrue the pension contributions and to discount the pension ben-

efits. In the deterministic setting, we have used the forward rates and have been

2.1. C APITAL CALCULATIONS 19

derived from the term structure of the spot interest rate. We have used the equation below to calculate the forward rates using the spot interest rate.

F _a+h =  (1 + r _a+h ) ^a+h (1 + r _h ) ^h



− 1 , (2.3)

where r is the spot interest rate, a is the time to maturity (in years), and h is the horizon (in years).

This forward rate can be interpreted as the spot interest rate h years into the future with a time to maturity a. Using this formula, we have filled a two dimensional forward rates matrix with the dimensions time to maturity and horizon. The time to maturity, as its name already suggest, is the number of years until the investment is settled.

The horizon, on the other hand, can be seen as moving the settlement date further into the future. So, for example when h = 43 and a = 1, it means that the investment is settled at the corresponding forward rate for a period of one year at the beginning of age 68. This is because the payments are made at the beginning of the year. As already mentioned before, the capital calculations start at the beginning of age 25.

Once the forward rates have been calculated, they are used to accrue the premiums and to discount the pension benefits as showed in Figure 2.2 and Figure 2.3. For every payment the time to maturity and horizon will be determined to see which forward rate is applicable.

After accruing the investment portfolio and discounting the benefits, it is possible to see what the total future value of the invested premiums and the present value of the pension benefits is at retirement age. This gives an indication about the coverage ratio. To recall, the coverage ratio is the relationship between the current available capital and the future pension obligations. Note that the term coverage ratio is used in our research loosely and does not correspond to the definition of coverage ratio used in the DB system. The coverage ratio says something about the relationship between the premiums and the expected pension benefit. We have used this ratio as a solvency measure and serves as an input in the utility function. In the following section we explain the relationship between the utility function and the coverage ratio in more detail. To compute the coverage ratio we have used the following equation:

x _t = I _t

B _t , (2.4)

where x is the coverage ratio, I is the value of the investment portfolio, and B is the

present value of the benefits at time t.

20 C HAPTER 2. A SSESSMENT FRAMEWORK

The investment portfolio in the formula above can be seen as the future value of the invested premiums (see the last column of Figure 2.1). The life cycle represents the percentage allocated to the return and matching portfolio and thus has an influence on the investment portfolio. In addition, we have incorporated the probability of sur- vival in the present value calculation of the benefits because there is a chance that the participant dies earlier. The pension benefits are first multiplied by the probability of survival before discounting. It is the probability of survival given that the partici- pant has reached the retirement age. This means that the pension benefit received at the beginning of 67 is multiplied by one because the probability of survival given that the participant reached the retirement age is one. The pension benefit at the beginning of 68 is multiplied by 0.9857 (mortality rate is 1.43% at the age of 67) because that is the probability the participant will receive the pension benefit. This is done up to the age of 103.

2.2 Measures

In economics, the utility function measures the welfare or satisfaction of a consumer as a function of consumption (Investopedia, 2018). In this case, the consumption is in terms of CR because it serves as a solvency indication of their pension. As we have already stated in the introduction chapter, we have used the Constant Relative Risk Aversion (CRRA) utility function, which is according to Yue (2014) one of the most commonly used utility functions in the pension industry. As the name already somewhat implies, risk aversion is the concept of human behaviour of disliking un- certainty. To give an example, if a player gets two options, a guaranteed payment of e50 or a 50% chance on e100 and 50% chance on e0, a highly risk-averse player will choose the guaranteed payment while the expected payouts are both the same.

A risk neutral player would be indifferent between the two options. Someone’s risk aversion is incorporated in terms of a risk aversion coefficient in the utility function.

The CRRA utility function is defined as follows:

U (x) = x ^1−γ

1 − γ , (2.5)

where x is the coverage ratio and γ is the risk aversion coefficient.

The risk aversion coefficient is an indicator of how much a person wants to avoid risk.

Not everyone is the same and therefore it is obvious that the risk aversion coefficient

varies per person. However, it is out of scope of our research to investigate what the

2.3. C ONCLUSION 21

real risk aversion coefficient is for everyone and to compute the utility function on an individual basis. Instead, we have created three profiles with different risk aversion coefficients, based on the paper of EDHEC (2014). We have used the following risk aversion profiles:

• Low risk aversion (offensive), γ = 2.

• Medium risk aversion (neutral), γ = 5.

• High risk aversion (defensive), γ = 10.

The utility values for every risk profile can be compared in order to determine the life cycle preferences. Interpreting one single utility is difficult because what does an utility of minus one, for example, mean? To get more feeling about the results we have used the certainty equivalent measure. This transforms a distribution of uncer- tain outcomes into a single value with probability one that has the same utility. It can be interpreted as a guaranteed CR that someone would accept rather than taking a chance on a higher, but uncertain, CR in the future. After determining the expected utilities it is straightforward to determine the certainty equivalent. Ranking alterna- tives by certainty equivalents is the same as ranking them by their expected utilities.

Rewriting Formula 2.5 results into the following certainty equivalent equation:

C = (E (U ) × (1 − γ))

, (2.6)

where E(U) is the expected utility and γ ∈ {2, 5, 10} is the risk aversion coefficient.

2.3 Conclusion

The capital calculations are indispensable because the pension benefit is not known

in advance in a DC pension plan. Therefore, we have started with the capital cal-

culations to get some insight in how the capital will develop over time and what the

pension benefit will be at retirement. In order to keep the connection with the most

used pension system, the DB pension plan, we have used a pension accrual per-

centage of 1.875% per year to construct the contribution table for the DC pension

plan. Together with the other stated assumptions, the CR is 100% when all the cap-

ital is allocated to the matching portfolio in the deterministic scenario. This results in

22 C HAPTER 2. A SSESSMENT FRAMEWORK

a pension benefit of e33,707.68. We have assumed that this is the lifetime annuity regardless of the fact that the asset allocation mix will change later in our research.

In addition, we have used the utility function to assess the trade-off between risk and return and to be able to compare the different life cycles fairly. We discuss the three existing unidirectional life cycles, which are tested later on in this research, in more detail in Section 4.1. They differ in terms of riskiness. Riskier life cycles might result in a higher pension benefit but at the same time have a greater chance on a terrible pension entitlement. In case a participant has already a good pension prospect then it might not be worth it to take extra risk to get an even better pension benefit. Besides that, not everyone is willing to take the same risk. Determining the risk aversion of every individual is not the goal of our research. Based on this rea- son, we have created three risk aversion profiles (defensive, neutral, and offensive) with their own risk aversion coefficient. The utility function is a useful measure for comparing the life cycles but the values do not say anything in itself. Therefore, we have transformed the utilities into certainty equivalents to be able to better interpret the results.

To conclude this chapter, the capital calculations and the utility function are essential

in order to assess the different life cycles. Together with the stochastic interest rates

and equity returns, which is the topic of the next chapter, it forms the basis to assess

the three existing unidirectional life cycles and the dynamic life cycle.

Chapter 3

Model building

We have built a simulation model to test the life cycles using stochastic interest rates and equity returns as input, which we discuss in this chapter. First, we give a short introduction about interest rates and equity returns. This emphasises the importance to use stochastic interest rates and equity returns. In Section 3.2 we explain the simulation model step by step, which consists of a dynamic Nelson- Siegel and a vector autoregression (VAR) model with a Markov regime switching component. The addition of a Markov regime switching component is not done a lot in the literature and therefore can be seen as an innovative element.

3.1 Interest rate and equity returns

The interest rate has a huge impact on the economy, and also on the pension indus- try. For the pension industry interest rates are used as input variable for calculating the values of the pension liabilities. Pensions are accrued over a long period and are usually paid for quite a long time. In determining the pension liabilities, pension funds must therefore use a long-term interest rate prescribed by the Dutch bank (DNB). Currently, the interest rates are historically low, around zero or even slightly negative. The interest rate drop is significant as can be seen in Figure 3.1. Be- cause of the low interest rate, pension funds are obliged to have more money in cash than, for example, a few years ago. This puts an enormous pressure on the pension industry and is also one of the reasons why the current pension industry is being discussed.

23

24 C HAPTER 3. M ODEL BUILDING

Figure 3.1: Historical EURIBOR interest rate percentages (Home Finance, 2019).

Because of the high importance of the interest rate a lot of research has been con- ducted to be able to model interest rate movements. These researches can es- sentially be classified into two frameworks. The first framework is to model the interest rate by modelling the evolution of the short rate. Under a short rate model, the stochastic state variable is taken to be the instantaneous spot rate (Wikipedia, 2019). From the short rate an entire yield curve is built up. If one chooses to fit the resulting interest rates on the current yield curve, the parameters of the model have to be calibrated to be consistent with the current observed prices of interest rate instruments. Well-known one-factor short rate models are for instance:

• Vasicek model (1977).

• Cox-Ingersoll-Ross model (1985).

• Ho-Lee model (1986).

• Hull-White model (1990).

One-factor short rate models are computationally efficient and are often analytically

and numerically tractable. The second class of interest rate frameworks are multi-

factor short rate models such as the Nelson and Siegel model (1987), the Longstaff-

Schwartz model (1992), and the Chen model (1996). The popular yield-curve repre-

sentation that was introduced by Nelson and Siegel is in 2006 extended by Diebold

and Li to a dynamic Nelson-Siegel model (DNS) and is used in our research to model

the interest rate. The main reason why we have chosen the Nelson-Siegel model is

because central banks extensively use this model to estimate the term structure of

3.1. I NTEREST RATE AND EQUITY RETURNS 25

interest rates (Bank for International Settlements, 2005). In addition, the model is parsimonious due to its simple functional form (see the formula in the next section) and can be extended to a no-arbitrage model.

In addition to the interest rate, modelling the equity returns is also an important part of this research. The return on equity investments comes in the form of dividend payments or capital gains from the increase in equity prices. A lot of research has been done regarding stock price modelling. Prices can fluctuate considerably as expectation regarding earnings growth and risk premium changes. As investors are faced with high risks they also demand higher returns on equity investments.

We have used the MSCI All Country World Index as a proxy for worldwide equities.

Figure 3.2 gives an overview of the annual total returns on equities. As can be seen from the figure below the returns fluctuate quite a lot, which illustrates the difficulty to predict the future returns. However, Sengupta (2004) wrote in his book: ”We talk about simulating stock prices only because future stock prices are uncertain (called stochastic), but we believe they follow, at least approximately, a set of rules that we can derive from historical data and our other knowledge of stock prices. This set of rules is called the model for stock prices”. We discuss our model used to ‘explain’

the interest rate and equity return paths in the next section.

Figure 3.2: Historical equity returns.

26 C HAPTER 3. M ODEL BUILDING

3.2 Simulation model

Simulations can be used to show the possible effects of alternative conditions but building a simulation model can be a complex thing to do. In the remaining of this section we explain our model step by step, see Figure 3.3 for an overview of the model. First, we explain the two underlying concepts, which are the dynamic Nelson- Siegel model and the Markov regime switching. Then we discuss some interim results, after which we explain the refinement and calibration of the model. We have used Matlab and Excel to build the simulation model. We do not discuss these codes in detail but they can be found in Appendix F.

Figure 3.3: Flowchart simulation model.

Dynamic Nelson-Siegel

A yield curve is a compilation of the interest rates with different times to maturity and

visually plotted as a curve. A three factor representation of the yield curve was first

introduced by Nelson and Siegel in 1987. Since then much research has been con-

ducted to improve the NS model. Well known is the extended model by Svensson

(1995). This model includes an extra factor to provide more flexibility. Another ex-

tension came from Diebold and Li (2005) who transformed the model into a dynamic

version making the parameters time-dependent. Once again, research have been

done to improve the DNS model. Examples of such extensions are models that

incorporate a time-varying loading parameter, volatility or unconditional mean. Fer-

guson and Raymar (1998) and Cairns and Pritchard (2001), however, showed that

the nonlinear estimators are extremely sensitive to the starting values used and that

the probability of getting local optima is high. Taking these drawbacks into account,

most researchers have fixed the loading parameter and have estimated a linearised

version of the Nelson-Siegel model (Annaert et al., 2012). Therefore, we have used

3.2. S IMULATION MODEL 27 the following DNS formula to model the yield curve:

y t (τ ) = β 1,t + β 2,t ×  1 − e ^−λ×τ λ × τ



+ β 3,t ×  1 − e ^−λ×τ

λ × τ − e ^−λ×τ



, (3.1)

where y t (τ ) is the yield with maturity τ at time t, λ is the loading parameter, and β 1 ,β 2

and β 3 are the level, slope and curvature factors respectively.

As can be seen in the formula above the loading parameter is not dependent on time, which simplifies the underlying assumptions of the model. The loading factor determines the exponential decay of the slope and curvature factor. The other vari- ables, the latent factors β ₁ , β ₂ , β ₃ , are dependent on time. These betas are called level, slope, and curvature respectively and carry some level of economical inter- pretation (Koopman et al., 2012). The first component equally influences the short and long-term interest rate and can therefore be interpreted as the overall level. The second component converges to one if the maturity goes to zero and converges to zero if maturity goes to infinity. This indicates that this component influences mainly the short term interest rate. The third component is associated with medium term interest rates because it is a concave function which converges to zero if maturity goes to zero and also converges to zero if maturity goes to infinity. The loading parameter influences the moment when the third component reaches its maximum.

Figure 3.4 gives an overview of the loading factors in relation to time to maturity.

These three loading factors together can capture a lot of different kind of shapes observed in yield data.

Figure 3.4: Factor loadings DNS model γ = 0.0609 (Diebold & Li, 2005).

Now that we have introduced the DNS model, the next step is to estimate the param-

eters. The first step is to find the optimised loading parameter. We have gathered

28 C HAPTER 3. M ODEL BUILDING

monthly zero coupon swap rates from January 2000 through March 2019 with dif- ferent time of maturities using Bloomberg. These maturities are 1 up to 10-year, 12, 15, 20, 25, 30, 40, and 50-year. We have estimated the loading parameter using the optimisation toolbox of Matlab and the Ordinary Least Squares (OLS) method.

OLS regression is a statistical method that estimates the relationship between a re- sponse variable and one or more explanatory variables. The method estimates the relationship by minimising the sum of the squares in the difference between the ob- served and predicted values of the response variable configured as a straight line, and is also referred as linear regression (Dickey et al., 2001). Initially, a loading pa- rameter is set in order to compute the error in the first iteration. In every iteration a loading parameter is chosen in order to compute the level, slope and curvature fac- tors. With these factors the swap rates are estimated which are then compared to the real data to calculate the error. For the next iteration the optimisation command in Matlab automatically chooses a new loading parameter and again computes the level, slope, and curvature factors to calculate the error. This process is repeated many times to find the optimal loading parameter which is the one with the minimum sum of squares. We have programmed this method in Matlab and the code can be found in Appendix F. The resulting loading parameter is 0.579, which we use in the remainder of our study. Finally, we have used this loading parameter to compute the time series of the level, slope and curvature factors. These time series are used to compute the parameters for the Markov regime switching.

Markov regime switching

Once the loading parameter, and the level, slope and curvature factors have been estimated we have extended the model with a Markov regime switching compo- nent. This model is widely applied in both finance and macroeconomics to incorpo- rate regime switches. Financial time series occasionally display dramatic breaks in their behaviour, for example due to a financial crisis or government policy changes.

Therefore it makes sense to incorporate regime switches in the DNS model. The

idea behind Markov regime switching is that processes can occur in different states,

or regimes. As the behaviour of the time series changes, regime switches are as-

signed to these changes, making the time series alternate between a predetermined

number of states. The goal of this step is to evaluate when a regime changes and

to estimate the values of the parameters associated with each regime. We have

used a Vector Auto Regression (VAR) model to compute the coefficients, the covari-

ance matrix, and the transition probability matrix. VAR is a commonly used model in

economic data analysis to simultaneously analyse multiple time series and capture

linear interdependencies. An advantage of a VAR model is that this model is easy

to use. In a VAR model each response variable has its own equation explaining the

evolution based on its own lagged values, the lagged values of other variables, and